The second link, in Spanish, mentions that the shares will be sold to an investment company controlled by Enrique and Ignacio Cueto, who control 25.7% of LAN and are the airline's CEO and President, respectively.

It seems as if instead of the Cueto's having to buy the 26.3% of LAN outright, they will buy a controlling stake (around 51%) in Pinera's Axxion holding company, which in turns owns 19.1% of LAN. This maneuver is legal for publicly-traded companies under Chilean law. Pinera could then sell the remaining stake in Axxion that the Cueto's don't buy to other investors in the open trading market, ridding himself of any impressions of conflict-of-interest that he would bring about if he kept ownership in Chile's largest airline.